Marker assisted selection using best linear unbiased prediction

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Marker assisted selection using best linear unbiased

prediction

R.L. Fernando, M. Grossman

To cite this version:

Original

article

Marker

assisted selection

_using

best

linear

unbiased

_prediction

R.L. Fernando

M.

Grossman

University of Idlinois, Department of Animal _Sciences, 1207 West _Gregory Drive _Urbana,

IL 61801, USA

(received

12 _{May 1989; accepted} 28 _August

₁₉₈₉₎

Summary - Best linear unbiased _prediction

_(BLUP)

is _applied to a mixed linear model

with additive effects for alleles at a market _quantitative trait locus

(MQTL)

and additive effects for alleles at the _{remaining quantitative} trait loci

_(QTL).

A recursive

_algorithm

is _developed to obtain the covariance matrix of the effects of _MQTL alleles. A simple

method is _presented to obtain its inverse. This approach allows simultaneous evaluation of fixed effects, effects of _{MQTL alleles,} and effects of alleles at the _{remaining QTLs, using} known _{relationships} and _phenotypic and marker information. The _approach is sufficiently

general

to accommodate individuals with partial or no marker information. Extension of

the _approach to BLUP with _multiple markers is discussed.

marker-assisted selection - best linear unbasied prediction - genetic marker

Résumé - Sélection assistée par un _{marqueur: utilisation du} meilleur _prédicteur

linéaire sans biais _(BLUP). La méthode du BLUP

(meilleure

prédiction linéaire sans

biais)

est _appliquée à un modèle linéaire mixte _comprenant des

effets

additifs associé aux allèles d’un locus _{quantitatif flanqué} d’un _gène marqueur, et d’effets

additifs

pour les autres locus _{quantitatifs.} Un algorithme récursif permet d’obtenir la matrice de covariances

associée aux

effets

des allèles du locus marqué. Une méthode simple est aussi proposée pour calculer l’inverse de cette matrice. Cette approche permet d’évaluer simultanément

les

_{effets fixés,}

les _effets des allèles du locus _marqué, et les

_{effets génétiques additifs}

de l’ensemble des autres _{locus, d’après} les relations de _parenté, les données _{phénotypiques}

et _{l’information} sur les _marqueurs. Cette _approche est assez _{générade pour tenir compte}

de données incomplètes chez certains individus. On discute l’extension à un BL UP avec

plusieurs marqueurs.

sélection assistée par un marqueur - meilleure prédiction linéaire sans _{biais - marqueur}

génétique

INTRODUCTION .

Genetic

_engineering

techniques

have

_produced

a

_variety

of molecular

_genetic

mark-ers with the

_potential

to

_identify

a

_large

number of

_{genetic polymorphisms}

_(Soller

*

and

Beckmann, 1982;

Smith and

_Simpson,

1986;

Schumm et

_al.,

_1988).

Marker-assisted selection is one

_application

of these

_techniques

to animal and

_plant

breeding.

Information on marker loci that are linked to

_quantitative

trait

_loci,

to-gether

with

_{phenotypic information,}

could be used to increase

_genetic

progress

by

increasing

accuracy of selection and

by reducing generation

interval

(Soller,

1978;

Smith and

_Simpson,

_1986).

Geldermann

₍₁₉₇₅₎

_proposed

a

least-squares procedure

to estimate effects of

marker alleles on

_quantitative

traits. Based on selection index

principles,

Soller

(1978)

combined marker information and

_phenotypic

information to obtain

_genetic

evaluations. This method has been used to

_study

the additional

_genetic

progress

expected

from marker-assisted selection

_(Soller,

1978;

Soller and

_{Beckmann, 1983,}

Smith and

_Simpson,

_1986).

Because of the

_complex

nature of animal

_breeding

data,

however,

these methods _{may not} be

_{applicable directly}

to marker-assisted selection with field data.

Data from field-recorded

_populations

are affected

_{by non-genetic}

nuisance

fac-tors, such as _age of

animal,

_age of

dam,

_{management system,} season of birth and

herb.

_Also,

non-random

_mating,

selection and

_{overlapping generations}

contribute to the

_complexity

of the data. Best linear unbasied

_prediction

_(BLUP;

Henderson,

1973, 1975,

1982)

deals with these

_{complications}

when

_{predicting breeding}

val-ues from

_phenotypic

data. The

_objective

of this paper is to present

methodology

for the

application

of BLUP to marker-assisted selection in animal

_breeding.

Each

methodological

development

is illustrated with a numerical

example using

a

_single

hypothetical pedigree

METHODOLOGY

Consider a

single polymorphic

marker locus

(ML),

closely

linked to a

_quantitative

trait locus

_(QTL).

Let

_MP

and

_Mi

_l

denote alleles at the ML that individual i inherited from its

_paternal

_(p)

and its maternal

_(m)

parent, and let

_QP

and

_Q7

denote alleles at the market

_QTL

_(MQTL)

linked to

_M!

and

_Mil,

as shown below:

Let

_vf

and

_vi&dquo;

be the additive effects of

_Q

_p

and

_Q7.

Additive effects of alleles at the

_{remaining QTLs,}

unlinked to the

_ML,

will be denoted

_by

the residual additive effect ui.

Now,

the additive effect for

individual i,

ai, can be written as

The usual model to obtain BLUP if additive

_{effects, given phenotypic information,}

is

where yiis the

phenotypic

value of

individual i,

xi

is a vector of known constants,

/

3

is a vector of unknown fixed

_effects,

and ei is a random error.

_Using

equ.(2),

BLUP

covariance matrix of ai values. Note that this covariance matrix

depends

on the

type of

_genetic

information available. When

_{only relationship}

information

_(r)

is

available,

the covariance of _ai values is

which is

_proportional

to the numerator

_relationship

matrix

_(e.g.,

_Henderson,

_1976).

When marker information

_(m)

is also

_available,

the covariance matrix ai values is

It can be shown that

_G

_alr

_i-

_,

_m

_,

_alr

_G

in

_general.

For

_example,

the covariance between half-sibs that receive the same ML allele from their common _parent is

_higher

than

the covariance between half-sibs that receive different ML alleles. This is because half-sibs

_receiving

the same ML allele also receive the same

MQTL

allele with

greater

frequency

than half-sibs

_receiving

different ML alleles. A. Marker model

I

To obtain BLUP with

_phenotypic

and marker

_information,

it is convenient to use

which is

_equivalent

to

_equ.(2).

The covariance matrix of vi values

(G&dquo;)

depends

on

relationship

and marker information. The covariance matrix of ui values

(G

u

)

depends only

on

_relationship

information and is

_proportional

to the numerator

relationship

matrix

_(e.g.,

Henderson,

1976).

Given the covariance matrices

G

v

and

G

u

,

BLUPs of vi and ui values can be obtained

_using

the mixed model

_equations

(Henderson, 1973).

The inverse of

_G

_u

_,

which is

_required

on the mixed model

equations,

usually

is obtained

_using

an

_{algorithm given by}

Henderson

_(1976).

A recursive

_algorithm

to construct

_G

_v

is

_given

in section

_B,

and an

algorithm

to

obtain its inverse is in section C.

B. Covariance matrix of

_MQTL

effects

l.

_Theory.

To construct

_G

_v

_,

consider the covariance between additive effects of

MQTL

alleles. Without loss of

_generality,

consider

_{only paternal MQTL}

alleles.

Suppose arbitrary

individuals o and o’ have sires s and s’. The

MQTL

alleles

inherited

_by

o and o’ from their sires are

QP

and

Q!,

_having

additive effects

_vP

and

V

’ .

For

_{paternal MQTL}

alleles in o and

o’,

the covariance between their additive

effects

_vo

_P

and

_vo,

is

where

_Var(vo)

=

w

is the additive variance of _an

MQTL

allele and

_P(Q!

_Q

_pt)

is the

_probability

that

_Qo

is identical

_by

descent to

_QP

_I

_.

For an

_arbitrary

_pair

of

individuals,

one is not a direct descendent of other. If o is not a direct descendant

of the

_o’,

_QP

can be identical

_by

descent to

_QP,

in 2

_mutually

exclusive ways:

1) Qo

is

identical

_by

descent to the maternal

_MQTL

allele of the sire of o’

_{(1!9, )}

and o’ inherits

_QP

_I

or

2) QP

is identical

_by

descent to the

paternal

MQTL

allele of the sire of o’

_and

If marker information is

_available,

the conditional

_probability

that o’ inherits

_Q!/,

given

that o’ inherits

_M

_:

_’,

is

_{(1 - r),}

where r is the recombination rate between the ML and the

_MQTL.

Thus if o’ inherits

_M

_:

_’,

the

_probability

in

_equ.(4)

can be

calculated

_recursively

as

Similarly,

if o’ inherits

_MS

If marker information is not

_available,

so that it is not known whether o’ inherits

M9

or

_M

_fi

_,

0.5

_replaces

r in

equs.(5)

and

(6).

This is

because,

in the absence of

marker

_information,

_Q!I

and

_Qfi

have

_{equal probability}

of

_being

transmitted to o’.

The above

_development

leads to a tabular method to construct

_G

_v

_,

which is similar to the method used to construct the numerator

_relationship

matrix

_(e.g.

Henderson,

1976).

Note that

G

v

has twice as many rows as individuals because each individual has 2 effects: 1 for the

_paternal

and 1 for the maternal

_MQTL

allele. The rows and columns of

G2!

should be ordered so that those

_{corresponding}

to progeny follow those for their parents. Let the row indices of

_G

_v

_,

_{corresponding}

to the effects of

_MQTL

alleles of individual

_{o( vg, v’),}

be

_{iP, io ;}

of its sire

_{s(vP, v7 ) ,}

be

_i!,i!;

and of its dam

_d(vd,

_!),

be

_id,

_i’.

Also,

let element

_{i j}

of

_G

_v

be _gij. Then from

_{equs.(4), (5)}

and

_(6),

the elements of row

_io,

below the

_diagonal,

are obtained as

for j

= 1...

io -

1,

where p

§

= r if o inherits

_M9

or

_pP,

=

(1 —

r)

if o inherits

_Mfi.

Elements of column

_iP,

above the

_diagonal,

are obtained from the

_{corresponding}

row elements because

G

v

is

_symmetric.

Similarly,

elements of row

_17,

below the

diagonal,

are obtained as

for

j

= 1...

io

-1

,

where

p

7

=_rif _oinherits

Md

and

_p

₇

=

(1 -

r

)

if o inherits

_{Md .}

Elements of column

_im,

above the

_diagonal,

are obtained from the

_{corresponding}

row elements.

From

_equ.(4),

the

_diagonal

elements of _Gv are

equal

to

_o,2.

If marker information

cannot be used to determine which of the 2 marker alleles o are inherited from its

sire or its

dam,

then 0.5

replaces

pP

in

equ.(7a)

or

_p

₇

in

_(7b).

2. Numerical

_example.

Consider the

_pedigree

in Table 1. To construct

_G

_v

_,

rows

and columns are

_{arranged by}

individual and

by

paternal

and maternal

MQTL

alleles

within individual

_{(Table II).}

For

convenience,

we will assume that

av =

1 and that r = 0.1. The first two individuals are assumed to be

_unrelated;

thus the upper left

4 x 4 submatrix of

G

v

is the

_identity

matrix. Elements on the

diagonal

are

equal

to

_or2=

1.

_Now,

row elements below the

diagonal

can be obtained from

equs.(7a)

and

_(7b);

column elements above the

_diagonal

are obtained

by

_symmetry. Each row

element for

_vl

_is

_equal

to

_{(1 - r)}

= 0.9 times the

corresponding

_row element for

vi

1

plus

r = 0.1 times the

corresponding

row element for

_{vi .}

Each row element for

v3

is

_equal

to r = 0.1 times the

corresponding

row element for

v2

_plus

(1 — r)

= 0.9

its sire is unknown.

_Thus,

each row element for

f!

_is

the mean

_(r

=

0.5)

of the

corresponding

row element for

vi

and for

_{vi .}

Marker information is available for

v4 ,

so that each row element for

_v4

is

(1 — r)

= 0.9 times the

corresponding

row

element for

_v3

_plus

r = 0.1 times the

_{corresponding}

_row element for

_{v3 .}

C.

_Algorithm

for

_inverting

G

v

1.

_Theory.

The

_approach

taken here follows that

_{by Quaas}

et al.

₍₁₉₈₄₎

and

_Quaas

(1988)

to invert the matrix of additive

_{relationships.}

We define a linear model to

relate the effect of the

_{paternal MQTL}

allele of an individual

(o)

to effects of

paternal

and maternal

_MQTL

alleles of its sire

_(s)

where

_EP

_is

a residual effect.

Similarly,

a linear model for effect of the maternal

MQTL

allele of o is

It can be shown that the residuals

_eP

in

_equ.(8a)

and

_E

_m

in

_(8b)

have a

_diagonal

where P is a matrix with each row

_containing

_only

two non-zero

_elements,

if the

parent is known or

_containing

only

zeros, if the parent is

_unknown;

and where is

a vector of residuals. For

_example,

_{row iP}

will have

_{(1 —}

_po)

in column

_iP

and

_pa

in column

_i

_l

_,

if the sire of i is known.

_Similarly,

row

₁₇

will have

(1 —

pl)

in the

column

_iP

and

_p7

in column

_{id ,}

if the dam of i is known.

To

_proceed,

we need the

_diagonal

elements of

G

e

.

Consider,

for

_example,

the

variance of

_eo.

From

_equ.(8a),

if the sire of o is known

because effects of

_MQTL

alleles of sire s are uncorrelated with residuals of its

offspring

o

_{(see Appendix).}

Hence

The covariance between the effects of

_paternal

and maternal

_MQTL

alleles can be

written as

where

F

S

is the

_inbreeding

of sire s.

Now,

equ.(10)

can be written as

because

_Var(vo)

=

Var(v§ )

=

Var(v7 )

=

Q

v,

and where

(1-

!)!

=

(1- r)r

for

_po

or for

_pg

=

(1 - r).

When the sire is not inbred:

_Var(eo)

=

2o!(l —

r)r,

if marker

information is

available;

or

_Var(e§)

=

a!/2,

if marker information is not available.

If the sire is not

_known,

_{Var(e§) =}

_w.

Similarly,

if dam of o is

_known,

the variance of

_e7

is

where

_{(1 -}

_{p7 )p§!}

=

(1 - r)r

for

p’

= r or for

_p-

=

(1 - r)

and where

F

d

is the

inbreeding

of dam d. When the dam is not inbred:

_{Var(eo )}

=

2o,2 (1 - r)r,

if marker

information is

_available;

or

Var(eo )

=

u§ /2,

if marker information is not available.

If the dam is not

_known,

_{Var(eo )}

=

Q

v.

Rearranging

(9),

v can be written as

for

_non-singular

_{(I -}

_P),

and thus

_G.&dquo;

can be written as

From

_equ.(14),

it is clear that

_a

_;

_;l

can be written as

As shown

earlier,

P has a

_simple

_structure, with each row

_containing

at most 2

non-zero

elements,

and

GE

is

diagonal.

To obtain the rules for

_inverting

_G

_v

₁

_,

_equ.(15)

is written as

where n is number of individuals in the

_pedigree,

_jq is

_{column j}

of

_Q,

and

_d

_j

is

diagonal element j

of

_{G§! .}

_By

definition of

_{Q, element j}

_{of qj} is

_{unity. Further,}

q

j will

have,

at most,

only

2 other non-zero

_elements;

_{for j = iP,}

element

_iP

_equals

- (1 - pP,)

and element

_is

_{equals -p}

_P

_o,

if the sire of o is known.

Similarly,

for

j

=

_i!,

element id

equals -(1 -

p!)

and element

_id

equals - p!,

if the dam

of o is known.

_{Thus, given}

_parent and marker information of an

individual,

the

contributions to

_G

_v

₁

_,

_{corresponding}

to effects of

_paternal

and maternal

_MQTL

alleles of the

_individual,

are

_easily

obtained.

Now,

to obtain the inverse of

G&dquo;:

1)

calculate

_diagonals

of

_{Gs :}

when the _parent is

_known,

the

_diagonal

is

_{given by}

_equ.(12a)

or

_(12b),

and when the _parent is

unknown,

the

_diagonal

is

_{o, V; 2}

₂₎

set

_v

_G

to

the null

matrix;

3)

for each

_offspring

o, with sire s and dam

_d,

add the

_following

to the indicated elements of

_G-

₁

_:

if sire is

known,

add

_{(1 -}

_p!)2di!

to

_diagonal

element

_{iP, iP;}

if dam is

_known,

add

_{(1 —}

_{p:;’ )}

₂

_d

_i:

_;.

to

_diagonal

element

_{il, il;}

.. - - 0 .... - -

d d

2. Numerical

_example.

Consider the

_pedigree

in Table 1. To construct

Go

we

again

take

_Q

_v

₌ 1 and r = 0.1. Because the _parents of

individuals

1 and 2 are not

known,

the first 4 elements on the

_diagonal

of

_G,

are

w

₌ 1. For individual

3,

each

parent is known and marker information is available.

_Thus,

from

_equs.(12a)

and

(12b),

the two

_diagonals

of

G,

corresponding

to effects of

_paternal

and maternal

MQTL

alleles of individual 3 are

2(1—r)r

= 0.18. Each _parent of individual 4 is also

known,

but the marker inherited from the sire is not known.

_Therefore,

the

_diagonal

of

G,

corresponding

to

_v4

_is

_0.5,

and that

_{corresponding}

to

_v4

is

_2(1—

_r)r

= 0.18.

The P matrix for this

_example

is

_given

in Table III. The first 4 rows of P are null

because parents of the first 2 individuals are not known. The sire of individual 3 is

_1,

and

_Mi

was transmitted to 3.

_Thus,

the row

_{corresponding}

to

_v3

_has

_{(1 —}

_r)

= 0.9

in the column

_{corresponding}

to

_vi

_and

r = 0.1 in the column

corresponding

to

vr. Similarly,

the dam of individual 3 is

_2,

and

_M2

was transmitted to 3.

Thus,

the row

_{corresponding}

to

_v3

has r = 0.1 in the column

corresponding

to

v2

and

(1 -

r)

= 0.9 in the column

corresponding

_to

v2 .

The sire of individual 4 is

1,

but

was transmitted to 4.

_Thus,

the row

_{corresponding}

to

_v4

has

_{(1 - r)}

= 0.9 in the

column

_{corresponding}

to

_vp

and r = 0.1 in the column

corresponding

_to

_v7n

The matrix

_Q

=

(I -

P’)

is

given

in Table IV. The

product

QG

E

1

Q’

is

given

in Table V. It can be verified that this is identical to the inverse of the matrix

G

v

D. BLUP with

_multiple

markers

If information on another marker locus linked to a

_QTL

is

_available,

the model can

be

_expanded

to include effects of alleles of this

_MQTL.

This

_{approach, however,}

results in 2n additional

_equations

for each marker introduced into the

_analysis.

Thus,

for a

_large

number of individuals

_(n)

and a

_large

number of

_{MQTLs, solving}

the mixed model

_equations

_may not be feasible. An alternative would be to use

equ.(2),

with

where

_v!i

and

_vk

_j

are effects of

paternal

and maternal alleles of the

k

t

i’

MQTL.

The covariance matrix of effects of

_MQTL

alleles at each locus

_{( Gv! )}

can be constructed

_using

the tabular method described in Section ILB.

Then,

assuming

gametic equilibrium,

the covariance of matrix _ai values

_{(Ga!T n!)}

can be obtained as

where Z is a n x 2n matrix with elements for row

_{i containing}

a 1

_{corresponding}

to

each of the

_paternal

and maternal

MQTL

effects of individual i and zeros for the

remaining

elements. The

_problem

with this

approach, however,

is that it could not

be

_applied

to

_large

_systems, unless a

_simple

_algorithm

to invert

_Galr,

_m

is available.

DISCUSSION

Results

_presented

here are an

_application

of BLUP to marker-assisted selection. This is a

generalization

of the method

presented by

Soller

(1978)

and Soller

and Beckmann

_(1983).

This

_{generalization}

allows simultaneous evaluation of fixed

effects,

MQTL

effects and the residual

_{QTL effects,}

using

known

_{relationships}

and

_phenotypic

and marker information. It is

_{sufficiently general}

to accommodate individuals with

_partial

or no marker information.

Several authors have calculated the additional

_genetic

_progress

_expected

from marker-assisted selection

_(Soller,

_1978,

Soller and

_{Beckmann, 1983;}

Smith and

Simpson,

1986).

Because the method

_presented

here is a

_{generalization}

of the method considered

_by

these

_authors,

their results

_give

an indication of the

advantage

expected by using

marker-assisted BLUP.

Application

of this

_procedure

_requires

_knowledge

of the recombination rate

_(r)

between the marker and the

_MQTL

and the variance of the additive effect of the

MQTL

alleles

_(a’).

_Assuming

that effects of

_MQTL

alleles are

_{normally distributed,}

the model

_presented

here could be used to estimate r and

_ufl

_by

restricted maximum likelihood

_(REML;

Patterson and

_Thompson,

_1971).

The robustness of REML

REFERENCES

Geldermann H.

₍₁₉₇₅₎

_{Investigation}

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characters in

animals

_by

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_Appl.

Genet.

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319-330

Henderson C.R.

₍₁₉₇₃₎

Sire evaluation and

_genetic

trends. In: Animal

_Breeding

and Genetics

_Symposium

in Honor

_{of Dr Jay}

L.

_Lush,

American

_Society

of Animal Science and American

_Dairy

Science

_Association,

_Champaign,

_IL,

_pp. 10-41 Henderson C.R.

₍₁₉₇₅₎

Best linear unbiased estimation and

_prediction

under a

selection model. Biometrics

_31,

423-439

Henderson C.R.

₍₁₉₇₆₎

A

_simple

method for

_computing

the inverse of a numerator

relationship

matrix used in

_prediction

of

_breeding

values. Biometrics

_32,

69-83 Henderson C.R.

₍₁₉₈₂₎

Best linear unbiased

prediction

in

_populations

that have

undergone

selection. In: World

_Congress

on

Sheep

and

Beef

Cattle

Breeding,

(Barton

R.A. & Smith W.C.

_ed.)

vol.

_1,

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_Press,

Palmerston

_{North, NZ,}

pp. 191-200

Henderson C.R.

₍₁₉₈₈₎

Use of an _average numerator

_relationship

matrix for

multiple-sire joining.

J. Anim. Sci.

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1614-1621

Patterson H.D. &

Thompson

R.

₍₁₉₇₁₎

Recovery

of inter-block information when block sizes are

unequal.

Biometrika

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545-554

Quaas

R.L.

₍₁₉₈₈₎

Additive

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model with _{groups and}

_{relationships.}

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Sci.

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1338-1345

Quaas

R.L.,

Anderson R.D. & Gilmour A.R.

₍₁₉₈₄₎

BLUP School

Handbook;

Use

of

Mixed Models

_for

Prediction and

_for

Estimation

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_(Co)variance

Components.

Animal Genetics and

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_Unit,

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of New

_England,

N.S.W.

2351,

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J.W.,

Knowlton

R.G.,

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Helms

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Hsiao

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Thurston J.G. & Donis-Keller H.

₍₁₉₈₈₎

Identification of more that 500 RFLPs

by

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₍₁₉₈₆₎

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Soller M.

₍₁₉₇₈₎

The use of loci associated with

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Soller

M.,

Beckmann J.S.

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APPENDIX

Proof that

_G,

is

_diagonal

Let o be an individual that is not a direct descendant of o’. From

_equs.(8a)

and

(8b),

the additive effects of the

_MQTL

alleles of o and o’ are

and

where z can take values p or

_m,

₌

s when z = _p,

_or

₌

d when z = _m.

Similarly,

z’ can take values _p or

m, !’

= s’ when z’ =

p, or

!’ =

d when z’ = m. Note

that for an

_arbitrary

_pair

of

individuals,

ones is not direct descendant of the other.

Therefore,

to _prove that

_G,

is

_diagonal,

it is sufficient to show thar the covariance between

_E

_’

and

_eo,

is null.

From

_equs.(A1)

and

_(A2),

the covariance between additive effects of

_MQTL

alleles

_v!

and

_v’,

can be written as

0 0

But,

from

_equs.(7a)

and

_(7b)

Thus,

for

_equ.(A3)

to

_equal

_(A4),

the third term in

_equ.(A3),

Cov(vz,

E

z,),

must be

zero. The same

_reasoning

can be used to show that

_{Cov( vf, E&dquo;!;)}

and

_{Cov(v!7ejl )}

are zero.

Therefore, given

that

Cov( v!,

E

z,), Cov( vf, and

Cov( vr

’

ejl )

are zero,

Cov(eo,

ejl )

must be zero.

Further, taking

o to be the a _parent of

o’,

the residual

(e

j

l

)

in

equ.(A2)

is

uncorrelated with

_vf,

and with

_vfi

in

_equ.(A2),

because

_{Cov(vz, E’O,)}

=

_0,

_as shown